Chevalley's Algebraic Geometry: Reminiscences & Picard Varieties

Transformation Groups, Vol. 4, No. 2-3, 1999, pp. 119-125 QBirkhs Boston (1999)
CHEVALLEY: SOME REMINISCENCES
C. S. SESHADRI
Chennai Mathematical
Institute,
92 G. N. Chetty Road
Chennai-600 017
India
[email protected]
I learnt my algebraic geometry first from Chevalley. I owe a lot to my
apprenticeship with Chevalley when I participated in his seminar on Picard
varieties in 1958-59.
1. In 1957 I had already spent four years as a research scholar at the
Tata Institute of Fundamental Research (TIFR). At the initiative of K.
Chandrasekharan who was in charge of its School of Mathematics at that
time, I was sent to work in Paris towards the beginning of the academic
year 1957-58. By that time I had submitted my thesis to the University of
Bombay. However, my training was not focussed and I had no idea as to
which area should be my speciality.
I looked forward to attending and imbibing the Cartan seminar. That
year it was on "Fonctions Automorphes". Several well-known mathemati-
cians were present at the seminar. The first talk was given by A. Weil.
Most of the mathematics went over my head, though I religiously continued
to attend the talks. That year the Chevalley Seminar was a continuation of
the one of the previous year on "Anneaux de Chow et Applications". Here
again I understood very little of what was talked about. One exception was
a course of Chevalley on "Fondements de la G@ometrie Alg~brique" (cf. [C1])
which I found that I could follow as it was self-contained. I concentrated
working on this course. Towards the end of 1957, I succeeded in proving
that a projective module over a polynomial ring in two variables is free. In
fact, if I remember right I related this to Serre when I met him for the first
time in December 1957. Thereafter, I had the benefit of talking to Serre on
several occasions. I followed his course on "Multiplicit@s et Intersections".
All these developments were encouraging but I was yet to find my major
field of research.
Chevalley announced a seminar on Picard Varieties in 1958-59 and at the
first meeting of the seminar H. Cartan chose me as one of the speakers. This
turned out to be a turning point in my career. The main part of the seminar
Received March 31, 1999. Accepted April 30, 1999.

120 C. S. SESHADRI
was devoted to the construction of the Picard variety of what is called a semi-
complete normal variety. I gave many talks following the personal notes of
Chevalley. Chevalley often invited me to his house for discussions concerning
the seminar. There I had the occasion to meet Mrs. Chevalley, as well as
his daughter. Sometimes I met also Japanese mathematicians with whom
Chevalley engaged in playing the game "Go".
Chevalley seemed quite pleased with my understanding and presentation
of his ideas in my talks, though they were certainly not models of pre-
sentation. In the course of the seminar, he suggested to me the problem
of constructing the Picard variety of an arbitrary complete variety (which
need not be normal). I solved this problem during the summer of 1959.
The seminar continued for a little while in 1959-60, where I presented my
solution. Then, for the written version of the seminar, he asked me to write
up my work. His construction of the Picard variety of a semi-complete nor-
mal variety, given at the seminar, was later published in his paper "Sur la
th~orie de la vari@t@ de Picard" (cf. [C2]).
It was indeed a great period to be in Paris. The Grothendieck revolu-
tion in algebraic geometry was just starting. I distinctly remember Serre
remarking during a chance meeting near the Pantheon sometime in the
beginning of 1958-59 that Grothendieck was planning to rewrite algebraic
geometry. Chevalley was one of the few among the older generation to un-
derstand and appreciate Grothendieck's work. Once he remarked to me that
Grothendieck had advanced algebraic geometry by fifty years and that he
understood the importance of the language of schemes because of some of
his own work. It is clear that he was referring to all his work suggesting the
possibility of constructing semi-simple group schemes over Z. One knows
that he did this later (cf. [C4]) and that this topic occupies an important
place in Grothendieck's SGA (Demazure-Grothendieck seminar).
The language of Chevalley's course "Fondement de la G6ometrie Alg@bri-
que" (cf. [C1]) and seminar on Picard varieties (cf. [SC2]) is, of course, of
the pre-Grothendieck era. In the "Fondement" the varieties are defined
over algebraically closed fields and there are no rationality questions. There
are no universal domains as in the Foundations of A. Weil and, in spirit,
the language is close to that of Serre's paper FAC (Faisceaux alg@briques
coh6rents), though there were no sheaves.
I can't refrain from narrating an amusing personal anecdote. During my
stay in Paris I was living at the Cit~ Universitaire at what is called the U.S.
house. During the last year of my stay I wanted to change to the German
house. For this I was asked to produce a letter from my "patron" to the
Director of the U.S. house. Presumably, it was a routine thing. I approached
Chevalley. He wrote such a good letter for me that the Director of the U.S.
house refused me permission to leave the house!
I returned to India by the end of the academic year 1959-60. On my way
back, I visited Florence, bearing in mind a remark or advice of Chevalley
that a visit to Florence is a must for understanding European culture. In

CHEVALLEY: SOME REMINISCENCES 121
1967 when I was returning from a visit to the U.S. I stopped in Paris.
Chevalley invited me and my wife to his house during this visit. Thereafter
I had very few contacts with Chevalley.
All through my association I could sense that Chevalley had great human
qualities. It has indeed been a great privilege to have worked with this great
man at close quarters.
2. A well-known structure theorem of Chevalley which he announced in 1953
states that if G is a group variety, there is a uniquely determined normal
connected affine (algebraic) subgroup such that G/H is an abelian variety.
His proof was based on the theory of Picard varieties but for quite sometime
he didn't publish his proof. The first published proofs of his theorem are
due (independently) to Barsotti and Rosenlicht (see the references in [C3]).
That Chevalley didn't publish his results soon may not come as a surprise,
as he was known not to hurry with his publications. There could have been
also another reason. The theory of Picard varieties was still evolving at
that time. For his proof he needed a more general theory of Picard varieties
than was available at that time. Chevalley also liked to understand and
present everything in his own way. All these he did subsequently in his
seminar on Picard varieties (cf. [SC2] and especially his paper [C2] which, as
I mentioned above, was the construction presented at the seminar) and the
paper [C3] giving the proof of his structure theorem (presumably on the lines
of his original proof). In fact, even his course "Fondements de la G@ometrie
Alg@brique" (loc. cit) seemed to be a part of his plan, as the language and
basic results of algebraic geometry used in his seminar and the papers are
based on his course and all these together constitute a self-contained entity.
Chevalley's proof of his structure theorem can be summarized as follows:
Recall that for a variety X, the Albanese variety Alb(X) is an abelian variety
which is universal for rational maps of X into abelian varieties. If, moreover
X is smooth, Alb(X) is universal for morphisms of X into abelian varieties.
Let G be a group variety (i.e., a connected, reduced algebraic group over
an algebraically closed field). Then the Albanese variety A = Alb(G) of
G is an abelian variety and one sees that the canonical map G ~ A is a
homomorphism of algebraic groups. Let H be the kernel of this homomor-
phism (with its reduced structure as a closed subgroup of G). Then it is
not difficult to perceive that Alb(H) is trivial, i.e., H has zero irregularity.
Then the crux of the proof is to show that if H is a group variety with zero
irregularity, then H is an affine algebraic group. To see this, recall that the
Picard variety of a variety X (if it exists) is the structure of a variety on
the set of divisor classes on X, which are algebraically equivalent to zero.
At that time it was known that the Picard variety of a smooth complete
variety X is the dual (i.e., the Picard variety) of the Albanese variety, a
discovery attributed to Chow. Then supposing that we have a theory of
Picard varieties for a good class of noncomplete varieties (H may not be
complete, in fact, a posteriori it is affine), it would follow that the Picard

122 C. S. SESHADRI
variety of H is trivial. Hence if D is an effective divisor on H and we denote
by Dh the translate of D by an element h of H, Dh is linearly equivalent to
D. Hence all Dh are in the complete linear system IDI. If H were complete,
this would be a projective space of dimension r. Let us pretend that this
is the case. Then through translations by elements of H, we could get a
homomorphism PD : H ~ PGL(r). Then by choosing a suitable finite
number of D's, we would get an injective homomorphism of H into a fi-
nite product of PGL(r), which would imply that H is affine, etc. To make
this intuitive proof work, one has therefore to work out a suitable theory of
Picard varieties for noncomplete varieties. Chevalley does this for what he
calls semi-complete normal varieties (a variety is semi-complete if the space
of global sections for any fractionary sheaf of ideals is finite dimensional).
He shows that the Picard variety of a semi-complete normal variety X is an
abelian variety and if, moreover X is smooth, it is the dual of its Albanese
variety (this is found in [C2] and was presented in the seminar [SC2]). The
group variety is not semi-complete but roughly speaking can be embedded
in one which is smooth and semi-complete and it turns out that this suffices.
3. I shall now give a brief idea of Chevalley's construction of Picard varieties
and the contents of his seminar on Picard varieties.
Due to the work of Cartier (whose thesis appeared around 1957-58), it
was realized that for general (i.e., not necessarily smooth) varieties, to have
a good theory of divisors one has to work with Cartier divisors (i.e., locally
principal divisors). Chevalley's construction of Picard varieties is based on
Cartier divisors. In the literature, the seminar on Picard varieties [C5] and
the paper [C2] are among the earliest to contain a systematic exposition
of Cartier divisors, e.g., families of Cartier divisors, functorial properties,
as well as some natural results on families of Cartier divisors (or divisor
classes) on complete varieties which are nowadays proved by appealing to
semi-continuity theorems on cohomology.
Let X be a variety with a family of Cartier divisor classes on X (or what
we would call nowadays a family of line bundles) parametrized by a variety
T (such that for a base point to E T, the corresponding line bundle on X is
trivial). Then the canonical set theoretic map T > Pic(X) is said to be
algebraic. If, moreover, T is a group variety and the map is a homomorphism
of (abstract) groups, the map is said to be an algebraic homomorphism.
According to Chevalley, the Picard variety of X is a universal object (if it
exists) for all algebraic homomorphisms. The modern definition of Picard
varieties is that of a universal object for all algebraic maps. Chevalley proves
that the Picard variety of X exists if it is semi-complete and normal; further,
it is an abelian variety and is a universal object for all algebraic maps.
Granting the existence of the Picard variety of X, then that it is universal
for all algebraic maps and also complete (hence an abelian variety), are
deduced from the following result. Let f : T ~ Pic(X) be an algebraic
map (with the condition about base points), T being a smooth curve but

CHEVALLEY: SOME REMINISCENCES 123
not necessarily complete. Then if J is the Jacobian of the complete curve
containing T and X : T > J the canonical map, f factors through an
algebraic homomorphism J ~ Pic(X). We see that this proof is close, in
spirit, to the modern one based on the valuative criterion for properness. A
systematic exposition of the construction of the Jacobian was given in the
seminar and is found in [C2].
The proof of the existence of the Picard variety of X (normal and semi-
complete) is motivated by the principle (mentioned above) that the Picard
variety is the dual of the Albanese variety. Chevalley proves the existence
of a "strict" Albanese variety for any variety X, i.e., there is an abelian
variety which has the universal property for morphisms from X into abelian
varieties, whereas, as we saw above, the classical Alb(X) is a universal object
for rational maps of X into abelian varieties. Then he shows the existence
of the Picard variety of an abelian variety A. For this he uses the fact
that there is a surjective map C ~ A, where C is a product of smooth
complete curves. One sees the existence of the Picard variety of C as an easy
consequence of that of the Picard variety (Jacobian) of a smooth complete
curve. Then the existence of the Picard variety of A follows by a general
argument. Finally, the Picard variety of the semi-complete normal variety
X is shown to be the Picard variety of the strict Albanese variety of X
(cf. [C2]).
As I mentioned above, Chevalley suggested to me the problem of con-
structing the Picard variety of a complete variety (not necessarily normal).
By that time one felt that using Cartier divisors (or line bundles) the earlier
theory for the case of smooth varieties would generalize to this case. In
the last expos6s of the seminar on Picard varieties (cf. [SC2]), Serre gave
a construction of the strict Albanese varieties (and some generalisations)
using Cartier's theory (Cartier operators etc.). Inspired by this, I found
that by Cartier's descent theory (i.e., descent theory for line bundles for
purely inseparable maps) one could prove the existence of the Picard vari-
ety of any complete variety X (in the sense of Chevalley, i.e., satisfying the
universal property for algebraic homomorphisms). I required, in addition,
the existence of the "field of rationality" for a Cartier divisor class on X.
This was proved by Chevalley (cf. expos6 7, [5]). To prove that the Picard
variety of X satisfies the universal property for all the algebraic maps re-
quired some effort. Let f : T > Pic(X) be an algebraic map such that
T is a smooth algebraic curve (and f(to) is trivial for some to C T). One
cannot expect that f would factorise through the Jacobian of the complete
curve containing T, as the Picard variety of X is, in general, not com-
plete. However, assuming that the Picard variety exists, one realises that
f has to factorize through some "generalized Jacobian of T" ( this is not
uniquely determined). This meant that one had to prove that the map f
has a "module" (see Serre's book [JPS]). Again, I profited by a remark of
Serre. It was realized around that time that Rosenlicht's generalized Jaco-
bian is, in fact, the Picard variety of a suitable complete nonnormal curve (cf.